Introduction to Algebraic Topology

Office hours: Monday, 2 pm-3 pm and by appointment

Grade distribution: 30% homeworks, 30% Midterm, 40 % Final

Midterm: Tuesday, March 8

Final exam: Take-home

TA: Kat Christianson

TA office hours: 4-5pm on Tuesdays and 12-1pm on Thursdays

Approximate Syllabus.

Part I. Basic homotopy theory

• Homotopy and homotopy equivalence.
• Fundamental group and higher homotopy groups.
• CW complexes.
• Calculation of the fundamental group, van Kampen theorem.
• Covering spaces and their classification.
• Relative homotopy groups, long exact sequences of pairs.
• Fibrations, long exact sequence of a fibration.
• Manifolds, degree of a map.

Part II. Homology

• Chain complexes, elements of homological algebra.
• Homology of CW complexes.
• Homotopy and homology, Hurewicz theorem.
• Homology and Cohomology.
• Kunneth formula.
• Applications.
• Manifolds, Poincare duality.

Lecture-to-lecture syllabus. Will be updated in timely manner (with notes).

• Lecture I. Basic topological notions. Homotopy and and Homotopy equivalence (pdf).
• Lecture II. Fundamental group and higher fundamental groups. Basics (pdf).
• Lecture III. Van Kampen theorem (pdf).
• Lecture IV. CW complexes (pdf).
• Lecture V. Covering spaces (pdf).
• Lecture VI. Covering spaces (continued) (pdf).
• Lecture VII. Relative homotopy groups (pdf).
• Lecture VIII. Fibrations, fiber bundles (pdf).
• Lecture IX. Singular homology (pdf).
• Lecture X. Singular homology (continued) (pdf).
• Lecture XI. Eilenberg-Steenrod axioms (pdf).
• Lecture XII. Cellular homology (pdf).
• Lecture XIII. Degree of a map and cellular homology (pdf).
• Lecture XIV. Cohomology (pdf).
• Lecture XV. Projective modules and ext. functor (pdf).
• Lectures XVI-XVII. Cohomology ring, continued Poincare duality (pdf).
• Lecture XVIII. Separation theorems (pdf).
• Lecture XIX. Extra techniques for homotopy groups (pdf).
• Lecture XX. Hurewicz map, further technique (pdf).
• Lecture XXI. Smooth manifolds (some notions) (pdf).
• Lecture XXII. Elements of Morse theory (pdf).
• Lecture XXIII. Morse theory (continued) (pdf).

Homeworks:

Recommended books:

• V. A. Vassiliev "Introduction to Topology"
  http://www.amazon.com/Introduction-Topology-Student-Mathematical-Library/dp/0821821628
• A. Hatcher, "Algebraic topology", (free online book, standard textbook)
  http://www.math.cornell.edu/~hatcher/AT/ATpage.html
• A.Fomenko, D.Fuchs, and V.Gutenmacher, "Homotopic Topology", (old russian textbook, also very useful)
  http://www.math.columbia.edu/~khovanov/algtop2013/Fuchs1.pdf